MEASURING THE SPIN C. Tonini A. Lapi P. Salucci OF SPIRAL GALAXIES

I N T R O D U C T I O N

Tonini, Lapi, Shankar & Salucci 2006, ApJ, 638, L13

The

mechanism of galaxy formation

involves the cooling and condensation of baryons inside the gravitational potential well provided by the Dark Matter (DM); in spirals, a rotationally supported disk is formed, whose structure is governed by angular momentum acquired through tidal interactions during the precollapse phase. The dynamics of the dark halo is directly related to the disk scale length (see Fall & Efstathiou 1980). This tight connection between halo dynamics and disk geometry is quantified by the spin parameter  .

In the present work, we propose to determine the angular momentum and the spin parameter of disk galaxies by making use of the observed matter distribution in spirals, and of observed scaling relations between halo and disk properties

.

To this purpose, we adopt a set of assumptions: (i) we use an empirical relation that links the disk mass to that of its DM halo; (ii) we suppose total specific angular momentum conservation during the disk formation, i.e.,

J B DM



M B M DM

in terms of the baryonic and halo masses and of the related total angular momenta; (iii) we assume two components for the baryonic matter, i.e. a centrifugally supported disk with an exponential surface density profile, and a gaseous (HI+He) component. Finally, we perform the computation for a Burkert (1995) halo.

A N G U L A R M O M E N T U M

For the DM halo of given virial mass, we adopt a Burkert distribution, parametrized by the density and mass profiles:  (

r

)   0 (

r



R

0 )(

R

0 3

r

2 

R

0 2 )

M h

(

r

)  4

M

0    ln   1 

r R

0    tan  1  

r R

0    1 2 ln   1 

r

2

R

0 2     

M

0  1 .

6  0

R

0 3 The fundamental parameters of the stellar disk and halo mass distributions can be obtained straightforwardly by means of observational scaling relations, linking the disk mass to the halo mass, to the halo central density, and to the disk scale length.

M D



Disk mass vs Halo mass

2 .

3  10 10

M

 1 ( 

M

(

H M H

/ 3 /  3 10 11 

M

10 11 

M

) 3 .

1  ) 2 .

2 (Shankar et al. 2006) derived by the statistical comparison of the galactic halo mass function with the stellar mass function; the related uncertainty is around 20 %, mainyl due to the mass-to-light ratio used to derive the stellar mass function from the galaxy luminosity function. The halo mass range is wide enough to include most of the spiral population, except dwarfs.

log(

R D

Disk scalelenght vs Disk mass /

kpc

)



0 .

633



0 .

379 log(

M D

/ 10

11

M



)



0 .

069



[log(

M D

/ 10

11

M



)]

2 inferred from dynamical mass determinations by Persic,Salucci & Stel (1996); consistent with the data by Simard et al. (1999). The disk is exponential with surface density:  (

r

)   0 exp( 

r

/

R D

) log( 

Halo central density vs Disk mass

0 /

g



cm

3 )   23 .

515  0 .

964 (

M D

/ 10 11

M

 ) 0 .

31 determined from the disk mass through the relation given by the Universal Rotation Curve (Burkert & Salucci 2000).

The core radius

R

is then obtained by requiring that the total 0 mass inside the virial radius equals the virial mass.

M gas

Gas mass vs Disk mass

 2 .

13  10 6

M

 

L B

10 6

L

  0 .

81 1   0 .

18 

L B

10 8

L

   0 .

4  from the disk luminosity (Persic & Salucci 1999). The gas is much more diffuse than the stars, reaching out several disk scale lengths (Corbelli & Salucci 2000; Dame 1993); since most of the angular momentum comes from material at large distances (van den Bosch et al. 2001), the gas can add a significant contribution to the total angular momentum, especially in small spirals where the gas to baryon fraction is close to 0.5.

The

disk angular momentum

is:

J D

 2    0 

D

(

r

)

rV rot

(

r

)

rdr



M D R D V halo f R

It depends linearly on the disk mass and radial extension; the DM distribution enters through the integrated velocity profile, and the shape factor

f R

(that varies slowly and at most by a factor 1.3 throughout our whole range of masses).

Here we show the specific angular momentum of the disk as a function of the rotation velocity at 2.2\,R_D. Left: the solid line is the result from this work, adopting the Burkert profile; the dashed line is the best-fit relation from the data collected by Navarro & Steinmetz (2000). Right: the Tully-Fisher relation; the solid line represents the result from this work and the dashed line illustrates the fit to the data by Giovanelli et al. (1997).

S P I N P A R A M E T E R

The

halo angular momentum

J H

 

J D

 The

spin parameter

J gas



M

is obtained as a function of the baryonic one:

M



H M D gas

(in its two common definitions here) is strictly related to both the dynamics and the geometry of the system:  

J E GM

1 / 2 5 / 2  ' 

J

2

M VIR V VIR

To compute the probability distributions

R VIR P

(  ) 

J halo

and 2 (

P M

(  '

VIR

)  

M D J D

 

M J gas gas

)

V VIR R VIR

, we use of the galactic halo mass function, fitted by the Schechter formula, in sampling the galactic masses for which we compute  and  ' . During this procedure we have convolved these values with a gaussian scatter of 0.15 dex that takes into account the statistical uncertainties in the empirical scaling laws we adopt; these are mostly due to the determination of

R D

.

As shown in the last figure (bottom panels), we find a distribution peaked around a value of about   0 .

03 and  '  0 .

025 , when the gas is considered. These values are close to the result of the simulations by D'Onghia \& Burkert (2004), who find  '  0 .

023 for spirals quietly evolving (experiencing no major mergers) since z=3. In addition, BD (2005) argue that this value provides a very good fit to the observed relation between the disk scale length and the maximum rotation velocity. We also stress that our most probable value of  is in agreement halos that evolve mainly through smooth accretion, hinting towards a quiet evolution for spirals (see paper for references).

The

gas distribution

was taken with a Gaussian profile (Corbelli & Salucci 2000) ; the detailed shape of the profile do not significantly affects its contribution to the angular momentum: the main factors are its mass and spatial extension.



gas

 [

M gas

 ( 2

k

1 2  2

k

2 2 )

R D

2 ]

e

 (

r

/

k

1

R D

)  (

r

/

k

2

R D

) 2

J gas

 4    0 

gas

(

r

)

rV rot

(

r

)

r

2

dr

C O N C L U S I O N S

We computed the angular momentum of systems DM halo + spiral galaxy, adopting observational scaling relations for the baryons and the Burkert profile for the DM; we find that the gas component adds a significant contribution. We obtain  '  0 .

025 and   0 .

03 2004) have shown that the distribution of the spin parameter for the whole halo catalogue peaks at around  Burkert (2004) highlight that if one restricts one's attention to halos that hosts spirals and have not experienced major mergers during the late stages of their evolution (z< 3), the average spin parameter turns out to be around  different evolutions in halos that have evolved mainly through major mergers or smooth accretion: in the former case values around 0.03, well in agreement with our result. Thus our findings point towards a scenario in which the late evolution of spiral galaxies may be characterized by a relatively poor history of major merging events.